Gorenstein Modules Induced by Foxby Equivalence

Let R , S be arbitrary associative rings and C a semidualizing (R,S) -bimodule. For a subcategory \mathscr{H} (resp. \mathscr{T} ) of the category of left R -modules (resp. left S -modules), we introduce \mathscr{H}_{C} -Gorenstein projective and flat modules (resp. \mathscr{T}_{C} -Gorenstein injective modules). Under certain conditions, we prove that the \mathscr{H}_{C} -Gorenstein projective dimension of any left R -module is at most n if and only if the projective dimension of any C -injective left S -module and the injective dimension of any module in \mathscr{H} are at most n . The dual result about the \mathscr{T}_{C} -Gorenstein injective dimension of modules also holds true. As a consequence, we get that the supremum of the C -Gorenstein projective dimensions of all left R -modules and that of the C -Gorenstein injective dimensions of all left S -modules are identical; and the maximum of the common value of the quantities and its symmetric common value is at least the supremum of the C -Gorenstein flat dimensions of all left R -modules. Moreover, we obtain some equivalent characterizations for the finiteness of the left and right injective dimensions of _{R}C_{S} in terms of the properties of the projective and injective dimensions of modules relative to various classes of C -Gorenstein modules. As an application, we provide some support for the Wakamatsu tilting conjecture.